Yes, of course they can and do. From the first conversation with my house bank, we received a commitment and agreed to sign as late as possible due to the 9-month interest-free period, so that it does not expire too early. And it was exactly right that way. However, it was clear from the beginning that the interest conditions apply from the day of signing and not from the initial conversation. But it was to our advantage because the interest rate went down even further ;)